The W-boson propagator

Every weak charged-current process — a neutron turning into a proton, a muon decaying, a quark changing flavour — runs through the exchange of a single virtual particle: the W boson. Its propagator is the internal line that carries momentum and spin from one weak vertex to the other, and it is what makes the weak interaction look weak. Structurally it is the massive cousin of the QED photon propagator: where the photon carries $-ig_{\mu\nu}/q^2$ , the W carries $\frac{-i}{q^2-m_W^2}\!\left(g_{\mu\nu}-q_\mu q_\nu/m_W^2\right)$ . Two changes are forced by the W's mass of about 80 GeV. The denominator gains a $-m_W^2$ , which pushes the pole away from $q^2=0$ and hands the force a finite range. The numerator gains a $q_\mu q_\nu/m_W^2$ piece, the fingerprint of the third, longitudinal polarisation that a massive spin-1 particle has and a photon does not. The single most important thing this object does is happen at low energy. In nearly every weak decay the momentum transfer is minuscule beside the W mass, $|q^2|\ll m_W^2$ , and the propagator freezes to the constant $ig_{\mu\nu}/m_W^2$ . The exchanged W can no longer be resolved; the interaction shrinks to a point, and we recover the four-fermion contact theory Fermi wrote down in 1934. Matching that limit fixes $G_F/\sqrt2=g_W^2/8m_W^2$ , and reading it correctly overturns a piece of folklore: the weak force is feeble not because its coupling is small — $g_W\approx0.65$ is larger than the electric charge — but because a heavy mediator sits in the denominator.

Key takeaways

The W propagator is $\frac{-i}{q^2-m_W^2}\!\left(g_{\mu\nu}-q_\mu q_\nu/m_W^2\right)$ — the photon propagator $-ig_{\mu\nu}/q^2$ plus a mass in the denominator and a longitudinal term in the numerator.
The $q_\mu q_\nu/m_W^2$ piece is the polarisation sum of a massive spin-1 boson, $\sum_\lambda\epsilon^{\lambda*}_\mu\epsilon^\lambda_\nu=-g_{\mu\nu}+q_\mu q_\nu/m_W^2$ ; contracted with light-fermion currents it is suppressed by $m_f^2/m_W^2$ and usually dropped.
For $|q^2|\ll m_W^2$ the propagator collapses to the constant $ig_{\mu\nu}/m_W^2$ — the exchange becomes point-like and reproduces Fermi's four-fermion contact interaction.
Matching that limit fixes $\dfrac{G_F}{\sqrt2}=\dfrac{g_W^2}{8m_W^2}$ , and $G_F=1.1663787\times10^{-5}\ \mathrm{GeV}^{-2}$ is read to ppm from the muon lifetime $1/\tau_\mu=G_F^2 m_\mu^5/192\pi^3$ .
Solving for the coupling gives $g_W\approx0.65$ and $\alpha_W=g_W^2/4\pi\approx1/30$ — larger than $\alpha_{\rm em}\approx1/137$ .
Key idea: the weak interaction is intrinsically not weak — its apparent feebleness at low energy is entirely due to the large W-boson mass in the propagator denominator, which suppresses every amplitude by $1/m_W^2$ .

The core insight

The weak interaction is intrinsically not weak — its apparent feebleness at low energy is entirely due to the large W-boson mass appearing in the propagator denominator, which suppresses the amplitude by $1/m_W^2$ .

Where this lives today

The two faces of the propagator — frozen plateau and resonant pole — are both front-line physics. On the plateau side, the muon lifetime is now measured to the part-per-million by the MuLan experiment at PSI, $\tau_\mu=2.1969811\ \mu\mathrm{s}$ , which fixes $G_F$ to $1.1663787\times10^{-5}\ \mathrm{GeV}^{-2}$ — one of the most precisely known constants in physics and, through equation (4), the anchor of the whole low-energy weak sector. On the pole side, the LHC produces W bosons in vast numbers precisely because $q^2$ there reaches and exceeds $m_W^2$ , where the propagator resonates rather than suppresses; the resulting Breit–Wigner lineshape, cut off by $\Gamma_W=2.085$ GeV, is a standard candle for detector calibration. The W mass extracted from these lineshapes is the propagator's most scrutinised parameter: the CDF collaboration's 2022 result, $m_W=80.4335\pm0.0094$ GeV, sits several standard deviations above the Standard-Model electroweak-fit prediction of about $80.357$ GeV — a tension in direct competition with ATLAS and CMS and one of the most-discussed hints of new physics. Whether or not it survives, every reading of it is a measurement of the pole in $1/(q^2-m_W^2)$ .

The physical picture

A weak decay is a two-vertex diagram stitched together by one internal W line, and that line is the propagator. At each end a fermion current couples to the W with strength $g_W/2\sqrt2$ ; in between, the propagator weights the exchange by how far off-shell the W is forced to be. Because the W is heavy, the amount it is off-shell in an ordinary decay is enormous — the momentum transfer is tiny beside $m_W^2$ — so the denominator is dominated by the mass and the whole exchange behaves as if it happened at one point. The first figure is the muon-decay diagram drawn twice: once with the resolved W line, once as the point-like contact interaction it becomes.

Figure

In muon decay the momentum transfer is q² ≈ m_μ² ≈ 0.011 GeV², utterly negligible beside m_W² ≈ 6460 GeV², so the propagator 1/(q²−m_W²) freezes to the constant −1/m_W² and the exchange looks point-like — Fermi's four-fermion contact interaction with strength G_F

The second figure is the propagator's own magnitude as a function of the energy scale it is probed at. Two regimes could not be more different. Below the W mass the curve is flat: the amplitude is stuck at $1/m_W^2\approx1.55\times10^{-4}\ \mathrm{GeV}^{-2}$ , independent of energy — this is the plateau that gives $G_F$ its constant value. As the momentum transfer climbs toward $m_W$ , the denominator $q^2-m_W^2$ passes through zero and the propagator blows up into a resonance, regulated only by the W's decay width $\Gamma_W\approx2.1$ GeV: this is a real W being produced, the physics of the CERN discovery and of every W at the LHC. Beyond the pole the mass becomes irrelevant and the curve falls as $1/q^2$ , exactly like a photon.

Figure

Magnitude of the W propagator, the size of 1/(q²−m_W²), versus the momentum-transfer scale √q² in GeV. Below m_W it is frozen at the plateau 1/m_W² = 1.55×10⁻⁴ GeV⁻²; at √q² = m_W = 80.4 GeV it resonates, its peak cut off by the width to about 1/(m_W Γ_W) ≈ 6×10⁻³ GeV⁻²; above the pole it falls as 1/q² like a massless photon

The core equations

The full Feynman rule for the internal W line, in unitary gauge, is the massive-vector propagator. The first two equations state it and explain the numerator; the rest extract the physics that dominates practice — the low-energy freeze and the constant it defines.

D_{\mu\nu}(q)=\frac{-i}{q^2-m_W^2}\left(g_{\mu\nu}-\frac{q_\mu q_\nu}{m_W^2}\right) \qquad (1)

where: $q$ — four-momentum flowing along the W line; $q^2$ — its invariant square (the momentum transfer); $m_W\approx80.4\ \mathrm{GeV}$ — the W mass; $g_{\mu\nu}$ — the metric. Reading it: the pole has moved from $q^2=0$ (photon) to $q^2=m_W^2$ , and the extra tensor structure $q_\mu q_\nu/m_W^2$ is the mark of a massive spin-1 field. The overall $-i$ and the sign of the denominator are the standard particle-physics conventions ( $+---$ metric). The numerator is not arbitrary — it is exactly the sum over the three physical polarisation states a massive spin-1 particle carries, evaluated on-shell. A photon has only two transverse states summing to $-g_{\mu\nu}$ ; the W's third, longitudinal state supplies the $q_\mu q_\nu/m_W^2$ .

\sum_{\lambda=1}^{3}\epsilon^{\lambda*}_\mu(q)\,\epsilon^{\lambda}_\nu(q)=-g_{\mu\nu}+\frac{q_\mu q_\nu}{m_W^2} \qquad (2)

where: $\epsilon^\lambda_\mu$ — the three polarisation vectors ( $\lambda=1,2$ transverse, $\lambda=3$ longitudinal); the relation holds for a real, on-shell W with $q^2=m_W^2$ . Reading it: the propagator numerator $(g_{\mu\nu}-q_\mu q_\nu/m_W^2)$ is minus this sum, which is why the propagator is said to "propagate three polarisations". The workhorse limit. In an ordinary weak decay $|q^2|$ is a fraction of a GeV², vanishing beside $m_W^2\approx6460\ \mathrm{GeV}^2$ , so the denominator is just $-m_W^2$ and every trace of momentum dependence disappears.

D_{\mu\nu}(q)\;\xrightarrow{\ |q^2|\ll m_W^2\ }\;\frac{i\,g_{\mu\nu}}{m_W^2} \qquad (3)

where: the longitudinal term is dropped because, contracted with light-fermion currents, it is suppressed by $m_f^2/m_W^2$ (Derivation C). Reading it: the amplitude no longer depends on how the energy is shared — the interaction has become point-like, a contact between four fermions. Matching the point-like amplitude to Fermi's phenomenological current-current theory fixes the relation between the fundamental coupling, the W mass, and the measured Fermi constant.

\frac{G_F}{\sqrt2}=\frac{g_W^2}{8\,m_W^2} \qquad (4)

where: $G_F=1.1663787\times10^{-5}\ \mathrm{GeV}^{-2}$ — the Fermi constant; $g_W$ — the SU(2) weak coupling; the factor $8$ comes from two vertices each carrying $g_W/2\sqrt2$ (Derivation B). Reading it: a single measured number, $G_F$ , ties together two theory quantities, and knowing $m_W$ turns $G_F$ into a measurement of $g_W$ . Finally, $G_F$ itself is not postulated — it is read off the muon lifetime, the cleanest weak decay there is, through the classic phase-space formula.

\frac{1}{\tau_\mu}=\frac{G_F^2\,m_\mu^5}{192\pi^3} \qquad (5)

where: $\tau_\mu$ — the muon mean lifetime; $m_\mu=105.66\ \mathrm{MeV}$ ; the $m_\mu^5$ is pure five-body phase space, and small $(m_e/m_\mu)$ and radiative corrections are understood. Reading it: measure $\tau_\mu$ , invert for $G_F$ , and equation (4) then delivers $g_W$ .

Derivation ABuilding the massive vector propagator from the Proca operator

The goal is to show that the W propagator (1) is forced — not guessed — by the requirement that it be the inverse of the free massive-vector wave operator, and to see precisely where the $q_\mu q_\nu/m_W^2$ term comes from. We start from the Proca Lagrangian, read off its momentum-space quadratic operator, invert it with a general tensor ansatz, and cross-check the answer against the polarisation sum (2) so that the counting of three physical states is manifest. A photon-style ansatz with only a $g_{\mu\nu}$ term will be seen to fail, which is the whole point.

Step 1. The free field is governed by the Proca Lagrangian $\mathcal{L}=-\tfrac14 F_{\mu\nu}F^{\mu\nu}+\tfrac12 m_W^2 A_\mu A^\mu$ , whose equation of motion in momentum space is $\big[-(q^2-m_W^2)g^{\mu\nu}+q^\mu q^\nu\big]A_\nu=0$ . Call the bracket $O^{\mu\nu}$ ; the propagator $D_{\nu\rho}$ is (up to an $i$ ) its inverse, defined by $O^{\mu\nu}D_{\nu\rho}=g^\mu{}_\rho$ . Why: a propagator is always the Green's function of the free operator — literally what you must convolve the source with to solve the wave equation.

Step 2. Because $O^{\mu\nu}$ is built only from $g^{\mu\nu}$ and $q^\mu q^\nu$ , so is its inverse: write $D_{\nu\rho}=A\,g_{\nu\rho}+B\,q_\nu q_\rho$ . Contracting $O^{\mu\nu}D_{\nu\rho}$ and demanding it equal $g^\mu{}_\rho$ gives $A=-1/(q^2-m_W^2)$ and, crucially, a $B$ fixed so the $q^\mu q_\rho$ terms cancel: $B=+1/[m_W^2(q^2-m_W^2)]$ . Why: the mass term is what makes $O^{\mu\nu}$ invertible at all — for a massless photon the operator is singular (gauge freedom), which is exactly why the photon propagator needs a gauge choice and the W's does not.

Step 3. Assembling, $D_{\mu\nu}=\dfrac{-1}{q^2-m_W^2}\big(g_{\mu\nu}-q_\mu q_\nu/m_W^2\big)$ ; restoring the standard $i$ gives equation (1). Direct check: with $P_{\nu\rho}=g_{\nu\rho}-q_\nu q_\rho/m_W^2$ , one finds $O^{\mu\nu}P_{\nu\rho}=-(q^2-m_W^2)\,g^\mu{}_\rho$ , the $q^\mu q_\rho$ pieces cancelling identically, so $P/[-(q^2-m_W^2)]$ is indeed the inverse. Why: this explicit cancellation is the algebraic heart of the result — the $-q_\mu q_\nu/m_W^2$ is precisely what the mass term demands, nothing added by hand.

Step 4. Cross-check against the physics of states. On-shell the numerator equals $-\sum_\lambda\epsilon^{\lambda*}_\mu\epsilon^\lambda_\nu$ of equation (2), summed over three polarisations. Setting $m_W\to0$ formally drops the third state and returns the transverse $-g_{\mu\nu}$ of the photon — but the $q_\mu q_\nu/m_W^2$ term diverges, the signal that a massive vector cannot be made massless by fiat; a Higgs mechanism is required to do it consistently. Why: the propagator "knows" the W has one more degree of freedom than the photon, and it refuses a smooth massless limit — foreshadowing electroweak symmetry breaking. *Verified: $O^{\mu\nu}P_{\nu\rho}=-(q^2-m_W^2)g^\mu{}_\rho$ by direct contraction; numerator $=-\sum_\lambda\epsilon\epsilon^*$ ; transverse part reduces to the photon numerator as $m_W\to0$ .*

Go deeper · Derivation B · 1 minThe low-energy limit and the Fermi constant

Derivation BThe low-energy limit and the Fermi constant

The aim is to reduce the full W-exchange amplitude to Fermi's four-fermion contact form in the regime $|q^2|\ll m_W^2$ and, by matching, to pin down the relation $G_F/\sqrt2=g_W^2/8m_W^2$ . We assemble the amplitude for a generic charged-current process from two Standard-Model vertices and one propagator, freeze the propagator to its low-energy constant, and read the coefficient of the resulting current-current product against the historical Fermi Lagrangian. The payoff is that a single measured number, $G_F$ , becomes a statement about $g_W$ and $m_W$ together.

Step 1. Each charged-current vertex contributes $\dfrac{-ig_W}{2\sqrt2}\gamma^\mu(1-\gamma^5)$ — the V−A structure with coupling $g_W/2\sqrt2$ . A process such as $\mu^-\to\nu_\mu\,e^-\bar\nu_e$ has two such vertices joined by the W propagator, so the amplitude is $\mathcal{M}=\Big[\dfrac{g_W}{2\sqrt2}\,j_1^\mu\Big]\dfrac{-g_{\mu\nu}}{q^2-m_W^2}\Big[\dfrac{g_W}{2\sqrt2}\,j_2^\nu\Big]$ with $j^\mu=\bar u\gamma^\mu(1-\gamma^5)u$ . Why: this is just the Feynman-rules bookkeeping — vertex, propagator, vertex — and the $q_\mu q_\nu$ term is already dropped by Derivation C.

Step 2. Freeze the propagator. For $|q^2|\ll m_W^2$ the denominator is $-m_W^2$ , so $-g_{\mu\nu}/(q^2-m_W^2)\to g_{\mu\nu}/m_W^2$ , and the amplitude becomes $\mathcal{M}\to\dfrac{g_W^2}{8m_W^2}\,j_1^\mu\,g_{\mu\nu}\,j_2^\nu=\dfrac{g_W^2}{8m_W^2}\,(j_1\!\cdot\! j_2)$ . Why: with the momentum dependence gone, the two currents simply contract — the exchange has become a point, and the amplitude is a pure current-current contact.

Step 3. Compare to Fermi. The phenomenological V−A contact Lagrangian is written $\mathcal{M}=\dfrac{G_F}{\sqrt2}\,(j_1\!\cdot\! j_2)$ , defining $G_F$ . Equating the two coefficients gives immediately $\dfrac{G_F}{\sqrt2}=\dfrac{g_W^2}{8m_W^2}$ , equation (4). Why: Fermi's constant was always an effective coupling standing in for "something heavy exchanged"; this line reveals what that something is and how strong its true coupling must be.

Step 4. Put in numbers. With $G_F=1.166\times10^{-5}\ \mathrm{GeV}^{-2}$ and $m_W=80.4$ GeV, $g_W^2=4\sqrt2\,G_F m_W^2=0.427$ , so $g_W=0.65$ and $\alpha_W=g_W^2/4\pi=1/29$ . Why: this is the punchline — $g_W$ is more than twice the electric charge $e=0.30$ , and $\alpha_W\approx1/30$ beats $\alpha_{\rm em}\approx1/137$ , so "weak" is a statement about $m_W$ , never about the coupling. *Verified: coefficient matching gives $G_F/\sqrt2=g_W^2/8m_W^2$ ; numerically $g_W=0.653$ , $\alpha_W=1/29.5$ , $\alpha_W/\alpha_{\rm em}=4.65$ .*

Go deeper · Derivation C · 1 minWhy the longitudinal term drops at leading order

Derivation CWhy the longitudinal term drops at leading order

The purpose here is to justify the everyday practice — used in the migrated article and in most textbook calculations — of replacing the full propagator (1) by the simpler $-ig_{\mu\nu}/(q^2-m_W^2)$ . The claim is that the $q_\mu q_\nu/m_W^2$ piece, once contracted with the external fermion currents that always flank a propagator, produces terms proportional to the tiny external fermion masses and is therefore doubly suppressed. We contract the term explicitly and use the Dirac equation to expose the mass factors.

Step 1. The offending term enters the amplitude as $\dfrac{q_\mu q_\nu}{m_W^2}\,j_1^\mu\,j_2^\nu$ , so it is enough to evaluate a single contraction $q_\mu j^\mu$ with $j^\mu=\bar u(p')\gamma^\mu(1-\gamma^5)u(p)$ and $q=p-p'$ the momentum the W carries between the fermions. Why: the whole term factorises into $(q_\mu j_1^\mu)(q_\nu j_2^\nu)$ , so if each factor is small the product is smaller still.

Step 2. Write $q_\mu j^\mu=\bar u(p')(\slashed p-\slashed p')(1-\gamma^5)u(p)$ . Using $\slashed p\,(1-\gamma^5)=(1+\gamma^5)\slashed p$ with the Dirac equations $\slashed p\,u(p)=m\,u(p)$ and $\bar u(p')\slashed p'=m'\,\bar u(p')$ , this becomes $\bar u(p')\big[m(1+\gamma^5)-m'(1-\gamma^5)\big]u(p)$ . Why: the derivative that $q_\mu$ represents acts on the external legs, and the equations of motion turn it into the on-shell masses $m,m'$ of those legs — the term literally measures how massive the participating fermions are.

Step 3. For the light fermions of ordinary weak decays — electrons, muons, light quarks — these masses obey $m,m'\ll m_W$ , so each factor $q_\mu j^\mu\sim m_f$ and the full term is of order $m_f^2/m_W^2$ relative to the leading contribution. In muon decay that ratio is $\sim(m_\mu/m_W)^2\approx1.7\times10^{-6}$ . Why: this quantifies "negligible" — the longitudinal correction is parts-per-million for muon decay, which is why the simplified propagator is used without apology. *Verified: $q_\mu j^\mu=\bar u(p')[m(1+\gamma^5)-m'(1-\gamma^5)]u(p)\propto m_f$ ; suppression $(m_\mu/m_W)^2=1.73\times10^{-6}$ for muon decay.*

Limiting cases

Case	Result	Meaning
Low momentum transfer, $\lvert q^2\rvert\ll m_W^2$	$D_{\mu\nu}\to i g_{\mu\nu}/m_W^2$	point-like contact; amplitude constant $\propto 1/m_W^2$ — Fermi theory, the regime of all ordinary weak decays
On the pole, $q^2\to m_W^2$	$q^2-m_W^2\to q^2-m_W^2+i\,m_W\Gamma_W$	a real, on-shell W is produced; the propagator becomes a Breit–Wigner resonance of width $\Gamma_W\approx2.1$ GeV
High momentum transfer, $q^2\gg m_W^2$	$D_{\mu\nu}\to -i g_{\mu\nu}/q^2$	the mass is irrelevant; the W propagates like a massless photon and unitarity is restored
Longitudinal term, $q_\mu q_\nu/m_W^2$	suppressed by $m_f^2/m_W^2$	drops against light external fermions; matters only for heavy legs (e.g. the top quark)
Massless limit, $m_W\to0$	transverse part $\to -ig_{\mu\nu}/q^2$	recovers the photon, but $q_\mu q_\nu/m_W^2$ diverges — a consistent massless limit needs the Higgs mechanism

Common trapthe weak force is fundamentally weaker than electromagnetism

This is the most common misreading of weak physics, and the propagator refutes it. The smallness of weak rates lives entirely in the factor $1/m_W^2$ from the frozen propagator, packaged into $G_F\sim g_W^2/m_W^2$ — not in the coupling $g_W$ . Solve equation (4) and the "weak" coupling comes out $g_W\approx0.65$ , larger than the electric charge $e\approx0.30$ , with $\alpha_W=g_W^2/4\pi\approx1/30$ actually exceeding $\alpha_{\rm em}\approx1/137$ . The interaction only looks feeble because at everyday energies the heavy W in the denominator crushes the amplitude. Raise the momentum transfer to $q^2\sim m_W^2$ — as colliders do — and the suppression evaporates: on the pole the "weak" interaction is anything but weak, producing real W bosons copiously. The lesson is that "weak" names an energy regime, not a strength.

Worked examples

1 · Extracting the true weak coupling (fully worked)

Given $G_F=1.166\times10^{-5}\ \mathrm{GeV}^{-2}$ and $m_W=80.4$ GeV, find the weak coupling $g_W$ and $\alpha_W=g_W^2/4\pi$ , and compare with electromagnetism.

Step 1. Rearrange the matching relation (4), $G_F/\sqrt2=g_W^2/8m_W^2$ , for the coupling: $g_W^2=8m_W^2\,G_F/\sqrt2=4\sqrt2\,G_F\,m_W^2$ .

Step 2. Insert numbers: $m_W^2=(80.4)^2=6464\ \mathrm{GeV}^2$ , so $g_W^2=4\times1.4142\times1.166\times10^{-5}\times6464=0.427$ , giving $g_W=\sqrt{0.427}=\mathbf{0.653}$ .

Step 3. Form the fine-structure analogue: $\alpha_W=g_W^2/4\pi=0.427/12.566=0.0339=\mathbf{1/29.5}$ . Compare with the electromagnetic $\alpha_{\rm em}=1/137$ and the charge $e=\sqrt{4\pi\alpha_{\rm em}}=0.303$ : the weak coupling is $g_W/e=2.16$ times larger, and $\alpha_W$ is $4.65$ times $\alpha_{\rm em}$ . The "weak" coupling is stronger than the electromagnetic one; the feebleness is all in $1/m_W^2$ .

2 · The Fermi constant from the muon lifetime (fully worked)

The muon has $m_\mu=105.66$ MeV. Use $1/\tau_\mu=G_F^2 m_\mu^5/192\pi^3$ with $G_F=1.1664\times10^{-5}\ \mathrm{GeV}^{-2}$ to predict $\tau_\mu$ , and compare with the measured value.

Step 1. Assemble the pieces in natural units: $G_F^2=1.360\times10^{-10}\ \mathrm{GeV}^{-4}$ ; $m_\mu^5=(0.10566)^5=1.317\times10^{-5}\ \mathrm{GeV}^5$ ; $192\pi^3=5953$ .

Step 2. The decay width is $\Gamma_\mu=\dfrac{G_F^2 m_\mu^5}{192\pi^3}=\dfrac{1.360\times10^{-10}\times1.317\times10^{-5}}{5953}=\mathbf{3.01\times10^{-19}\ \mathrm{GeV}}$ .

Step 3. Convert to a lifetime with $\hbar=6.582\times10^{-25}\ \mathrm{GeV\,s}$ : $\tau_\mu=\hbar/\Gamma_\mu=6.582\times10^{-25}/3.01\times10^{-19}=\mathbf{2.19\ \mu s}$ . The measured muon lifetime is $2.1970\ \mu s$ ; the tree-level formula lands within half a percent, the small remainder being the $(m_e/m_\mu)$ phase-space and QED radiative corrections. A microsecond lifetime, reproduced from $G_F$ alone.

3 · The range of the weak force (fully worked)

A heavy mediator gives a short-ranged force with range $R=\hbar/(m_W c)$ . Compute $R$ for the W and compare it with the size of a proton.

Step 1. In convenient units $\hbar c=0.1973\ \mathrm{GeV\,fm}$ , so the range is $R=\hbar c/(m_W c^2)=0.1973/80.4$ .

Step 2. Evaluate: $R=2.45\times10^{-3}\ \mathrm{fm}=\mathbf{2.45\times10^{-18}\ m}$ .

Step 3. Compare with the proton's charge radius $\approx0.84$ fm: the weak force reaches only $R/0.84\approx1/340$ of a proton radius. The W's 80-GeV mass confines the charged-current interaction to a few attometres — which is exactly why, at nuclear and atomic scales, it behaves as a zero-range contact interaction.

Physics of the portrait

The W propagator is the modern resolution of a puzzle Enrico Fermi posed in 1934: he described beta decay as a contact interaction of four fermions at a single point, with one empirical constant setting its strength. That worked beautifully at low energy and failed at high energy — the contact theory's amplitudes grow without bound and violate unitarity. The cure was to spread the point back out into the exchange of a heavy mediator, whose propagator supplies the missing $1/q^2$ falloff at high energy and reproduces Fermi's constant at low energy. Fermi's four-fermion vertex is quite literally the $|q^2|\ll m_W^2$ limit of this article's central object.

Origins · 1 minEnrico Fermi and how this was discovered

Enrico Fermi

1901–1954 · the four-fermion contact theory of beta decay

Enrico Fermi, working in Rome in the winter of 1933–34, wrote down the first quantitative theory of the weak interaction. Borrowing the structure of quantum electrodynamics — a current of the decaying nucleon coupled to a current of the emitted electron and neutrino — he proposed that beta decay proceeds through a four-fermion contact interaction at a single space-time point, its strength set by one constant now written $G_F$ and carrying his name. The paper was so far ahead of its evidence that Nature rejected it as "too remote from reality"; it was published in Italian and German instead, and it stood as the working theory of the weak force for a generation. Fermi's contact interaction correctly gave the shape of the beta spectrum and the lifetimes of nuclei, but it hid a flaw he could not have seen: as a genuine zero-range interaction, its cross-sections rise without limit with energy and eventually break the probabilistic consistency of the theory. The resolution, completed by Glashow, Weinberg and Salam three decades later, was that the "point" is not a point at all but the exchange of a very heavy charged boson — the W — whose propagator is this article's subject. In the low-energy limit that propagator collapses back to Fermi's contact vertex, and his constant reappears as $G_F/\sqrt2=g_W^2/8m_W^2$ . Fermi did not live to see the W: he died in 1954, twenty-nine years before it was found at CERN with almost exactly the mass his effective theory implied.

History

1934

— Fermi publishes his four-fermion contact theory of beta decay, defining the constant G_F

1957

— Sudarshan and Marshak propose the V−A structure of the charged weak current

1961

— Glashow introduces the SU(2)×U(1) gauge structure of the electroweak interaction

1967

— Weinberg formulates the electroweak model, giving the W and Z mass by spontaneous symmetry breaking

1983

— the UA1 and UA2 experiments at CERN's SppS collider discover the W boson near 80 GeV

Going further

Texts. Thomson, Modern Particle Physics, §11.5 (the W-boson propagator) and §11.5.1 (Fermi theory), with the massive spin-1 polarisation sum in Appendix D · Halzen & Martin, Quarks and Leptons, §12 (the weak interaction and intermediate vector bosons) · Griffiths, Introduction to Elementary Particles, 2nd ed., §9–10 · Peskin & Schroeder, An Introduction to Quantum Field Theory, §20.2 (spontaneously broken gauge theory and the massive vector propagator) · Schwartz, Quantum Field Theory and the Standard Model, §29 (electroweak theory).

Provenance. The propagator's massive-vector form and its low-energy freeze are grounded in Thomson §11.5, Eqs. (11.9)–(11.16); the numerator's polarisation-sum origin follows Thomson Appendix D. A verifier should confirm: (1) $D_{\mu\nu}=\frac{-i}{q^2-m_W^2}(g_{\mu\nu}-q_\mu q_\nu/m_W^2)$ is the inverse of the Proca operator; (2) the low- $q^2$ limit gives $ig_{\mu\nu}/m_W^2$ and hence $G_F/\sqrt2=g_W^2/8m_W^2$ ; (3) $g_W=0.653$ , $\alpha_W=1/29.5$ from $G_F$ and $m_W$ ; (4) the tree muon lifetime $\tau_\mu=2.19\ \mu s$ from $1/\tau_\mu=G_F^2 m_\mu^5/192\pi^3$ ; (5) the longitudinal-term suppression $(m_\mu/m_W)^2=1.7\times10^{-6}$ ; (6) the range $R=\hbar c/m_W=2.45\times10^{-3}$ fm.

Prerequisites. The Photon Propagator · The Weak Charged Current · The V−A Interaction · Virtual Particles and Feynman Diagrams. Leads to. The Fermi Constant · Muon Decay · Properties of the W Boson · Electroweak Unification.

Ask the tutor: Ask me to walk through how the same $1/(q^2-m_W^2)$ that makes muon decay take microseconds turns into a resonance when the LHC drives $q^2$ up to $m_W^2$ .

Why does the W propagator have q² − m_W² in the denominator while the photon has only q²?

The denominator of a propagator is $q^2-m^2$ for any particle of mass $m$ ; it vanishes — the propagator "goes on shell" — when the exchanged quantum can be a real particle. For the massless photon that happens at $q^2=0$ , so the electromagnetic force is long-ranged and its propagator never freezes to a constant. For the W, $m=m_W\approx80$ GeV, so the pole sits far out at $q^2=m_W^2$ and, for any everyday momentum transfer, the denominator is just $-m_W^2$ . That single difference — a mass in the denominator — is what turns a long-range $1/q^2$ force into a short-range, apparently feeble contact interaction. It is the whole story of why the weak force is weak.

What is the q_μ q_ν/m_W² term, and when can I ignore it?

It is the imprint of the W's third polarisation state. A massless photon has only two transverse polarisations, summing to $-g_{\mu\nu}$ ; a massive spin-1 boson has an additional longitudinal state, and the polarisation sum becomes $-g_{\mu\nu}+q_\mu q_\nu/m_W^2$ , whose second piece appears in the propagator numerator. You can ignore it at leading order whenever the fermions attached to the W are light: contracting $q_\mu$ with an external current gives a factor of that fermion's mass (via the Dirac equation), so the term is suppressed by $m_f^2/m_W^2$ — about a part in a million for muon decay. It must be kept when a heavy fermion such as the top quark sits on the line, and in careful higher-order or gauge-dependent calculations.

If g_W isn't small, why do weak decays take so long — the muon lives 2.2 microseconds?

Because rates depend on the amplitude squared, and every low-energy weak amplitude carries the frozen propagator factor $1/m_W^2$ . A muon-decay rate scales as $G_F^2\sim(g_W^2/m_W^2)^2$ , so even though $g_W\approx0.65$ is a healthy coupling, the two powers of $1/m_W^2$ with $m_W=80$ GeV crush the rate. The phase-space formula $1/\tau_\mu=G_F^2 m_\mu^5/192\pi^3$ turns that suppression into a width of $\sim3\times10^{-19}$ GeV, a lifetime of microseconds — astronomically long by particle-physics standards, but entirely due to $m_W$ , not to a small coupling.

What happens exactly at the pole q² = m_W² — doesn't the propagator blow up?

It would, if the W were stable. A real W is unstable, decaying with width $\Gamma_W=2.085$ GeV, and including that decay shifts the pole off the real axis: the denominator becomes $q^2-m_W^2+i\,m_W\Gamma_W$ . The magnitude of the propagator then peaks at a finite value $1/(m_W\Gamma_W)\approx6\times10^{-3}\ \mathrm{GeV}^{-2}$ rather than diverging, and the lineshape is a Breit–Wigner resonance of width $\Gamma_W$ . Physically the "pole" is simply the energy at which you produce a real W instead of a virtual one — this is exactly what happens in $W$ production at the LHC, and fitting the resonance is how $m_W$ and $\Gamma_W$ are measured.

Is the low-energy Fermi contact theory "wrong"?

No — it is the correct effective theory at low energy, just incomplete. Fermi's four-fermion interaction is precisely the $|q^2|\ll m_W^2$ limit of W exchange, and for nuclear beta decay or muon decay it is essentially exact, the corrections being of order $q^2/m_W^2\sim10^{-6}$ . What it cannot do is extrapolate to high energy: as a genuine contact interaction its amplitudes grow like $G_F\,s$ and violate unitarity around a few hundred GeV. Restoring the full propagator cures this — the $1/(q^2-m_W^2)$ falloff tames the growth — which is historically how physicists knew a heavy mediator had to exist long before the W was seen.

How is G_F actually measured?

From the muon lifetime, because muon decay is the cleanest weak process available: a single charged-current amplitude, no hadrons, well-understood radiative corrections. One measures $\tau_\mu$ extremely precisely — the MuLan experiment reached parts per million, $\tau_\mu=2.1969811\ \mu s$ — and inverts the theoretical relation $1/\tau_\mu=(G_F^2 m_\mu^5/192\pi^3)\times[1+\text{corrections}]$ . This yields $G_F=1.1663787\times10^{-5}\ \mathrm{GeV}^{-2}$ . Combined with $m_W$ from collider lineshapes, equation (4) then delivers $g_W$ , closing the loop between a stopwatch on muons and the fundamental strength of the weak interaction.

Why does a heavy mediator make a force short-ranged?

Because the range of a force is inversely proportional to the mediator's mass: $R=\hbar/(m c)$ , the Compton wavelength of the exchanged quantum. A massless photon has infinite Compton wavelength, so electromagnetism reaches to infinity as $1/r^2$ . The W, at 80 GeV, has a Compton wavelength of just $2.5\times10^{-18}$ m — about one three-hundredth of a proton radius — so the interaction it mediates looks, at any larger scale, like a zero-range contact. The same fact is the Yukawa potential $e^{-m_W r}/r$ that a massive-boson propagator Fourier-transforms into: the exponential kills the force beyond $R=1/m_W$ .